MS Comprehensive Exam: ALGEBRA
Instructions: Please work at most one problem on each side of each sheet of paper, for
ease in grading. Thanks. Also, don’t quote theorems identical to what you are asked to prove,
if there are any such theorems.
(1) Work just one of the following:
(a) If the order |G| of the finite group G is even, then there exists at least one element
a ∈ G such that a 6= e and a = a−1 .
(b) Suppose G is a group of order |G| = 2p with p a prime.. Show that G must have
a normal subgroup H of index 2. Recall that the index [G : H] is the number of
left cosets {xH : x ∈ G}.
(2) Given the ring Z[i] = {a + bi : a, b ∈ Z}, suppose
Ip := {a + bi ∈ Z[i] : p|a and p|b}, where i denotes the usual imaginary element
i2 = −1 in C.
(a) Show that I3 is a maximal ideal.
(b) Show that I5 is not a maximal ideal. Hint: Define M = h2 + ii to be the ideal
generated by (2 + i) in Z[i].
(3) Show that the following polynomials are irreducible over the given field F.
(a) x2 + 7 over R.
(b) x3 − 3x + 3 over Q.
(c) x2 + x + 1 over Z2 .
(d) x2 + 1 over Z19 .
(e) x3 − 9 over Z13 .
(f) x4 + 2x2 + 2 over Q.
(4) Suppose that F is a subfield of K with [K : F] = p, p a prime number.Show that
K = F(a) for every element a ∈ K that is not in F.